■ A right angle is an angle that measures exactly 90 degrees. A right angle is sumbolized by a square at the vertex. ■ An obtuse angle is an angle that measures more than 90 degrees, but less than 180 degrees. ■ A straight angle is an angle that measures 180 degrees. Thus, both its sides form a line. COMPLEMENTARY ANGLES Two angles are complimentary if the sum of their measures is equal to 90 degrees. 1 2 ∠1 + m∠2 = 90° Complementary Angles m Straight Angle 180° Obtuse Angle Right Angle Symbol – THE GRE QUANTITATIVE SECTION– 180 SUPPLEMENTARY ANGLES Two angles are supplementary if the sum of their measures is equal to 180 degrees. ADJACENT ANGLES Adjacent angles have the same vertex, share a side, and do not overlap. The sum of all possible adjacent angles around the same vertex is equal to 360 degrees. ANGLES OF INTERSECTING LINES When two lines intersect, vertical angles are formed. Vertical angles have equal measures and are supple- mentary to adjacent angles. 1 2 3 4 ∠1 + m∠2 + m∠3 + m∠4 = 360° m 1 2 ∠1 and ∠2 are adjacent Adjacent Angles 1 2 ∠1 + m∠2 = 180° Supplementary Angles m – THE GRE QUANTITATIVE SECTION– 181 ■ mЄ1 = mЄ3 and mЄ2 = mЄ4 ■ mЄ1 + mЄ2 = 180 ° and mЄ2 + mЄ3 = 180 ° ■ mЄ3 + mЄ4 = 180 ° and mЄ1 + mЄ4 = 180 ° BISECTING ANGLES AND LINE SEGMENTS Both angles and lines are said to be bisected when divided into two parts with equal measures. Example: Therefore, line segment AB is bisected at point C. According to the figure, ЄA is bisected by ray AC. 35° 35° A C A CB bisector 1 2 3 4 – THE GRE QUANTITATIVE SECTION– 182 ANGLES FORMED BY PARALLEL LINES When two parallel lines are intersected by a third line, or transversal, vertical angles are formed. ■ Of these vertical angles, four will be equal and acute, and four will be equal and obtuse. The exception to this is if the transversal is perpendicular to the parallel lines. In this case, each of the angles formed measures 90 degrees. ■ Any combination of an acute and obtuse angle will be supplementary. In the above figure: ■ Єb, Єc, Єf, and Єg are all acute and equal. ■ Єa, Єd, Єe, and Єh are all obtuse and equal. Some examples: mЄb + mЄd = 180 º mЄc + mЄe = 180 º mЄf + mЄh = 180 º mЄg + mЄa = 180 º Example: In the following figure, if m ʈ n and a ʈ b, what is the value of x? x ° (x + 10)° b m n a ab e f cd h g – THE GRE QUANTITATIVE SECTION– 183 Solution: Because Єx is acute, you know that it can be added to x + 10 to equal 180. Thus, the equation is x + x + 10 = 180. Solve for x:2x + 10 = 180 –10 –10 2x = 170 ᎏ 2 2 x ᎏ = ᎏ 17 2 0 ᎏ x =85 Therefore, mЄx = 85 and the obtuse angle is equal to 180 – 85 = 95. ANGLES OF A TRIANGLE The measures of the three angles in a triangle always equal 180 degrees. EXTERIOR ANGLES An exterior angle can be formed by extending a side from any of the three vertices of a triangle. Here are some rules for working with exterior angles: ■ An exterior angle and interior angle that share the same vertex are supplementary. ■ An exterior angle is equal to the sum of the nonadjacent interior angles. B A C 1 ∠4 + m∠3 = 180° and m∠4 = m∠2 + m∠1 2 3 4 m B AC m∠1 + m∠2 + m∠3 = 180° 3 2 1 – THE GRE QUANTITATIVE SECTION– 184 Example: mЄ1 = mЄ3 + mЄ5 mЄ4 = mЄ2 + mЄ5 mЄ6 = mЄ3 + mЄ2 ■ The sum of the exterior angles of a triangle equal to 360 degrees. Triangles More geometry questions on the GRE pertain to triangles than to any other topic. The following topics cover the information you will need to apply when solving triangle problems. CLASSIFYING TRIANGLES It is possible to classify triangles into three categories based on the number of equal sides: Scalene Triangle: no equal sides Isosceles Triangle: at least two equal sides Equilateral Triangle: all sides equal Scalene Isosceles Equilateral 3 4 6 5 1 2 – THE GRE QUANTITATIVE SECTION– 185 . Angle Right Angle Symbol – THE GRE QUANTITATIVE SECTION 180 SUPPLEMENTARY ANGLES Two angles are supplementary if the sum of their measures is equal to 180 degrees. ADJACENT ANGLES Adjacent angles have the same. that measures 180 degrees. Thus, both its sides form a line. COMPLEMENTARY ANGLES Two angles are complimentary if the sum of their measures is equal to 90 degrees. 1 2 ∠1 + m∠2 = 90 ° Complementary Angles m Straight. that measures exactly 90 degrees. A right angle is sumbolized by a square at the vertex. ■ An obtuse angle is an angle that measures more than 90 degrees, but less than 180 degrees. ■ A straight